How to compute $\mathbb{P}[X>0>Y, X/Y \leq -a]$ where $X$ and $Y$ are two independent standard normal random variables?

Question

Suppose $X$ and $Y$ are two independent standard normal random variables. I know that $X/Y$ has Cauchy distribution. But how to compute $$\mathbb{P}[X>0>Y, X/Y \leq -a]$$ where $a$ is a positive constant. (I don't think $X>0>Y$ and $X/Y \leq -a$ are independent, so the probability above cannot be factorized as product.)

Answer 1

Another way to compute this is to note that the 2d standard normal distribution is radially symmetric. Thus we can write $(X,Y) = (R\cos\theta, R \sin \theta)$ where $\theta$ is uniformly distributed on $[0,2\pi]$.

If you draw a picture of your problem, then you'll notice that in these polar coordinates, you're simply computing the probability that $\theta$ lies in between two fixed values, which is easily computed in terms of arctangent.